# Analysis Seminar

Friday, July 15, 2016 — 2:30 PM EDT

Michael Dritschel, University of Newcastle

“A Fej ́er-Riesz theorem for non-negative operator valued trigonometric polynomials in two variables ”

The classical Fej ́er-Riesz theorem states that any non-negative trigonometric polynomial of degree d with complex coefficients is the “square” of an analytic polynomial of the same degree, and this polynomial may be chosen to be outer. In the 70s, Marvin Rosenblum showed that the same theorem holds true for non-negative polynomials with coefficients which are bounded operators on a Hilbert space. In the 90s, I found a new proof of this theorem using Schur complements, and a later paper with Hugo Woerdeman improved the method for outer decompositions. The Schur complement method also works for strictly positive polynomials in several variables, though there is no control over the degree and number of analytic poly- nomials needed in the decomposition the closer the polynomial comes to not being strictly positive. A recent theorem of Claus Scheiderer implies, among other things, that complex valued non-negative trigonometric polynomials in two variables have sums of squares decom- positions in terms of analytic polynomials. Furthermore, his work also implies that for three or more variables, there will always exist non-negative polynomials without sums of squares decompositions. The methods used are from real algebra and have no obvious generalisation beyond the scalar setting. Using a refinement of the Schur complement technique, we show how to get the two variable result for operator valued trigonometric polynomials, as well as discussing why things break down for more than two variables.

MC 5403

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